Foundation Quantum Ledger Empty Ledger Balance

The empty ledger's balance

In the Recognition Science framework, a ledger is a discrete record of recognition events, each event carrying a cost. The framework models physical structure from the forced cost of these recognitions. A foundational declaration, empty_ledger_balance, states that the empty ledger, one with no entries, has a balance of zero. This is a formal theorem in the framework's machine-checked library, proved by definition: the ledger's balance is defined as the sum of the log-ratios of its entries, and the empty sum is zero.

The balance itself is defined as the sum of the log-ratios of the ledger's entries. Each entry has a positive ratio, and the balance is the sum of the natural logarithms of these ratios. For the empty ledger, there are no entries, so the sum is empty and equals zero. This is not an assumption or a postulate; it is a direct consequence of the definitions. The declaration is a theorem, tagged THEOREM, with the anchor IndisputableMonolith.Foundation.QuantumLedger.empty_ledger_balance.

This fact is a basis for the framework's treatment of quantum states as superpositions over ledger configurations. The framework proves that the total balance is conserved under updates: adding a pair of reciprocal entries, with ratios r and 1/r, leaves the balance unchanged because log(r) + log(1/r) = 0. The empty ledger's zero balance is the base case for this conservation. It establishes a consistent starting point: the empty ledger has no cost and no balance, and the framework's evolution preserves this balance.

What the declaration does not claim is equally precise. It does not claim that the empty ledger is the only ledger with zero balance; other ledgers with entries whose log-ratios sum to zero also have zero balance. It does not claim that the empty ledger has any physical existence or that it represents a vacuum state. It is a formal statement about the framework's definitions, not a physical claim. The declaration is a theorem, not a model choice, and its proof is by reflexivity, meaning it follows directly from the definitions of the ledger and its balance.

theorem empty_ledger_balance : emptyLedger.balance = 0 := rfl

/-- A ledger is a collection of entries with conservation constraint. -/
structure Ledger where
  /-- The entries in the ledger -/
  entries : List LedgerEntry
  /-- Total balance (sum of log-ratios) -/
  balance : ℝ
  /-- Balance equals sum of log-ratios -/
  balance_eq : balance = (entries.map (fun e => Real.log e.ratio)).sum

/-- **CONSERVATION THEOREM**: Applying an update preserves balance. -/
theorem ledger_balance_conserved (L : Ledger) (u : LedgerUpdate) :
    (applyUpdate L u).balance = L.balance := rfl

What this page does not claim

The empty ledger is the only ledger with zero balance. The empty ledger represents a physical vacuum state. The declaration establishes any property of ledgers with entries.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/QuantumLedger.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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